Searcharxiv⌕ Search

arXiv subjects

Madhuparna Das

Publications and source records attributed to Madhuparna Das.

14 recordsLinked to original sources
arXiv2026

Mapping Mathematical Hardness: Machine-Assisted Conjecture Discovery and the Quantification of Non-Triviality

Machine-assisted mathematical discovery has been a long-standing challenge in machine learning and artificial intelligence. In recent years, we have seen tremendous progress with generative AI, yet its contribution to automated discovery in advanced mathematical research has been limited. One of the most difficult benchmarks in this context is the Birch test, which asks whether a machine can discover truly novel and non-trivial mathematical structures without human intervention. In this work, we particularly focus on the branch of automated conjecture discovery. We use HypothesiX, an automated conjecture mining agent and analyse its generated conjectures related to the distribution of twin primes to verify the conditions of the Birch test. Furthermore, note that automated discovery is now operating at scale, but verifying its non-triviality still depends on human evaluation. We propose a benchmark to quantify the non-triviality of machine-generated conjectures using the Mahalanobis distance within an embedding cluster of selected known mathematical conjectures. We also note that this quantified benchmark can be used as an error indication signal to localise the incorrectness of a new mathematical statement, which autoformalisers fail to verify due to their limitations in proof discovery capability.

math.GM↗
arXiv2026

A central limit theorem for partitions involving generalised divisor functions

We define an $f$-restricted partition $p_f(n,k)$ of fixed length $k$ given by the bivariate generating series \begin{align*} Q_f(z,u) \coloneqq 1+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} p_f(n,k) u^kz^n =\prod_{k=1}^{\infty}(1+uz^k)^{Δ_f(k)}, \end{align*} where $Δ_f(n)=f(n+1)-f(n)$. In this article, we establish a central limit theorem for the number of summands in such partitions when $f(n)=σ_r(n)$ denotes the generalised divisor function, defined as $σ_r(n)=\sum_{d|n}d^r$ for integer $r\geq 2$. This can be considered as a generalisation of the work of Lipnik, Madritsch, and Tichy, who previously studied this problem for $f(n)=\lfloor{n}^α\rfloor$ with $0<α<1$. A key element of our proof relies on the analytic behaviour of the Dirichlet series \begin{align*} \sum_{n=1}^{\infty}\frac{σ_r(n+1)}{n^s}, \end{align*} for $\mathrm{Re}(s)>1$. We study this problem employing the identity involving the Ramanujan sum. Furthermore, we analyse the Euler product arising from the above Dirichlet series by adopting the argument of Alkan, Ledoan and Zaharescu.

math.NT↗
arXiv2024

Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions

In this article, we consider the weighted partition function $p_f(n)$ given by the generating series $\sum_{n=1}^{\infty} p_f(n)z^n = \prod_{n\in\mathbb{N}^{*}}(1-z^n)^{-f(n)}$, where we restrict the class of weight functions to strongly additive functions. Originally proposed in a paper by Yang, this problem was further examined by Debruyne and Tenenbaum for weight functions taking positive integer values. We establish an asymptotic formula for this generating series in a broader context, which notably can be used for the class of multiplicative functions. Moreover, we employ a classical result by Montgomery-Vaughan to estimate exponential sums with additive coefficients, supported on minor arcs.

math.NT↗
arXiv2024

Walking through the Gaussian Primes

The Gaussian Moat problem asks whether one can walk to infinity in the Gaussian integers using the Gaussian primes as stepping stones and taking bounded length steps or not. In this paper, we have analyzed the Gaussian primes and also developed an algorithm to find the primes on the $\mathbb{R}^2$ plane which will help us to calculate the moat for higher value. We have also reduced a lot of computation with this algorithm to find the Gaussian prime though their distribution on the $\mathbb{R}^2$ plane is not so regular. A moat of value $\sqrt{26}$ is already an existing result done by Genther et.al. The focus of the problem is to show that primes are getting lesser as we are approaching infinity. We have shown this result with the help of our algorithm. We have calculated the moat and also calculated the time complexity of our algorithm and compared it with Genther-Wagon-Wick's algorithm. As a new ingredient, we have defined the notion of primality for the plane $\mathbb{R}^3$ and proposed a problem on it.

math.NT↗
arXiv2024

Generalization of Bertrand's Postulate for Gaussian Primes

Bertrand's Postulate states about the prime distribution for the real numbers. The generalization of Bertrand's Postulate was proved by Das et al. [Arxiv 2018]. In this paper, we have formalized this idea for the Gaussian primes (or the primes on the complex plane). This result gives information about the prime distribution on the complex plane.

math.NT↗
arXiv2024

On the Extension of the Gaussian Moat Problem

In this paper, we have developed an algorithm for the prime searching in $\mathbb{R}^3$. This problem was proposed by M. Das [Arxiv,2019]. This paper is an extension of her work. As we know the distribution of primes will get more irregular as we are going to infinity and going to the higher dimensions. We have also shown that why it is not possible to extend the Gaussian Moat problem for the higher dimensions (more than four dimensional plane).

math.NT↗
arXiv2024

A Note on The Gaussian Moat Problem

The Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. In this paper, we have proved that the answer is `No', that is an infinite sequence of distinct Gaussian prime numbers can not be bounded by an absolute constant, for the Gaussian primes $p=a^2+b^2$ with $a,b\neq0$. We consider each prime $(a,b)$ as a lattice point on the complex plane and use their properties to prove the main result.

math.NT↗
arXiv2023

Partitions into semiprimes

Let $\mathbb{P}$ denote the set of primes and $\mathcal{N}\subset \mathbb{N}$ be a set with arbitrary weights attached to its elements. Set $\mathfrak{p}_{\mathcal{N}}(n)$ to be the restricted partition function which counts partitions of $n$ with all its parts lying in $\mathcal{N}$. By employing a suitable variation of the Hardy-Littlewood circle method we provide the asymptotic formula of $\mathfrak{p}_{\mathcal{N}}(n)$ for the set of semiprimes $\mathcal{N} = \{p_1 p_2 : p_1, p_2 \in \mathbb{P}\}$ in different set-ups (counting factors, repeating the count of factors, and different factors). In order to deal with the minor arc, we investigate a double Weyl sum over prime products and find its corresponding bound thereby extending some of the results of Vinogradov on partitions. We also describe a methodology to find the asymptotic partition $\mathfrak{p}_{\mathcal{N}}(n)$ for general weighted sets $\mathcal{N}$ by assigning different strategies for the major, non-principal major, and minor arcs. Our result is contextualized alongside other recent results in partition asymptotics.

math.NT↗
arXiv2019

Analysis of the Game "2048" and its Generalization in Higher Dimensions

We theoretically analyze the popular mobile app game `2048' for the first time in $n$-dimensional space. We show that one can reach the maximum value $2^{n_1n_2+1}$ and $2^{\left({\prod_{i=1}^{d} n_i}\right)+1}$ for the two dimensional $n_1\times n_2$ board and $d$ dimensional $n_1\times n_2\times \ldots \times n_d$ board respectively. We also present a strategy for the computer and a winning strategy for the human player in certain conditions.

cs.DM↗
arXiv2019

Revisiting Generalized Bertand's Postulate and Prime Gaps

It is a well-known fact that for any natural number $n$, there always exists a prime in $[n, 2n]$. Our aim in this note is to generalize this result to $[n, kn]$. A lower as well as an upper bound on the number of primes in $[n, kn]$ were conjectured by Mitra et al. [Arxiv 2009]. In 2016, Christian Axler provided a proof of the lower bound which is valid only when $n$ is greater than a very large threshold. In this paper, after almost a decade, we for the first time provide a direct proof of the lower bound that holds for all $n \geq 2$. Further, we show that the upper bound is a consequence of Firoozbakht's conjecture. Finally, we also prove a stronger version of the bounded gaps between primes.

math.NT↗